# ECE503

Probability and Random Processes for Engineering Applications
Fall and Spring
Catalog Data:

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ECE 503

Probability and Random Processes for Engineering Applications

Credits: 3.00

UA Catalog Description:  http://catalog.arizona.edu/allcats.html

Course Assessment:

Homework:  8 – 10 assignments

Exams:  2 Midterm Exams, 1 Comprehensive Final Exam

Typically: 10% Homework,

30% Exam #1,

30% Exam #2,

30% Final Exam.

Course Summary:

Graduate level treatment of probability, random variables, stochastic processes, correlation functions and spectra with applications to communications, control, and computers.

Prerequisite(s):
SIE230 or Equivalent
Textbook(s):

“Probability, Random Variables and Stochastic Processes,” 4th Edition by A. Papoulis and S.U. Pillai, McGraw-Hill, 2002. Other editions are OK.

Course Topics:

1.     Brief Overview of Logic/Proof Techniques.

2.     Probability Space – sample space, sigma field, probability measure.

3.     Conditional Probability.

4.     Total Probability.

5.     Bayes.

6.     Independence.

7.     Cartesian (combined) Experiments.

8.     One Random Variable.

a. Definition

b. Distribution/Density

c. Conditional Distributions/Densities

d. Total Prob/Bayes

e. Conditioning on zero probability events

9.     Functions of one RV.

a. Distributions/Densities

10.  Expectation.

a. Moments

b. Characteristic functions

c. Moment generating function

11.  Two Random Variables.

a. Distribution/Density

b. Joint/marginal

c. Independence

d. One function of two RVs

e. Two functions of two RVs

f. Moments

g. Covariance/Correlation/Orthogonal
h. Joint Characteristic Functions

i. Conditional Distributions/Moments

j. Min MSE Estimation

12.  N Random Variables.

a. Distribution/Density

b. Independence

c. N functions of N RVs

d. Covariance/Correlation Matrix

e. Characteristic Function

13.  Convergence of Random Sequences.

a. Everywhere, Almost Everywhere, in Probability, in MSE, in Distribution

b. Weak law of large numbers

c. Central limit theorem

14.  Random Processes.

a. Definition, Interpretation, Examples

b. 1st order, 2nd order, nth order, strict sense, wide sense

c. Moments-Mean, Variance, Covariance, Correlation

d. Properties

15.  Stationary – 1st order, 2nd order, nth order, strict sense, wide sense.

16.  Transformations of processes.

a. (focus on Linear Time Invariant Filter)
b. Covariance/Cross Covariance input/output

c. Power Spectral Density

d. Average Power

17.  Poisson Process.

a. Poisson Points

b. Time to first point (exponential)

c. Time to nth point (Erlang)

d. Time between points (exponential)

18.  Ergodicity.

a. Mean ergordic

b. Covariance ergodic

19.  Markov Processes.

a. Definition

b. Focus on Discrete Time

c. Transition probabilities

d. State occupancy probabilities

e. Homogeneous (time-invariant)

f. Transition Diagram

g. Transition Matrix

h. Irreducible

i. Aperiodic

j. Stationary Distribution

Class/Laboratory Schedule:

Lecture:  150 minutes/week

Prepared by:
Michael Marcellin
Prepared Date:
April 2013

University of Arizona College of Engineering